Cable impedance

This document tries to clear out some details of transmission lines
and cable inductance. This document is only a brief introduction to
those topics.
If you expect to work much with transmission lines, coaxial or
otherwise, then it will be worth your while to get a book on that
subject. The ideal book depends on your background in phsics or electrical
engineering, and in mathematics.

What is the cable impedance and when it is needed?

The basic idea is that a conductor at RF frequencies no longer behaves like a
regular old wire. As the length of the conductor (wire) approaches about 1/10
the wavelength of the signal it is carrying - good ol' fashioned circuit
analysis rules don't apply anymore. This is the point where things like
cable impedance and transmission line theory enter the picture.

The key tenet of all transmission line theory is that the source impedance
must be equal to the load impedance in order to achieve maximum power transfer
and minimum signal reflection at the destination. In real world case
this generally means that the source impedance is the same as cable impedance
and the value of the receiver in another end of the cable has also the
same impedance.

How cable impedance is defined ?

Characteristic impedance of the cable ratio of
the electric field strength to the magnetic field strength for waves
propagating in the cable (Volts/m / Amps/m = Ohms).

Ohm's Law states that if a voltage (E) is applied to a pair of terminals and a current (I) is measured in this circuit, the following equation can be used to determine the magnitude of the impedance (Z). The following formula
will hold truth:

Z = E / I

This relationship holds true whether talking about direct current (DC) or alternating current (AC).

Characteristic Impedance and is usually designated Zo or "Zed nought".
When the cable is carrying RF power, without standing waves, Zo also equals
the ratio of the voltage across the line to the current in flowing in the
line conductors. So the characteristic impedance is defined
with the formula:

Zo = E / I

The voltages and currents depend on the inductive reactance and
capacitive reactance in the cable. So the characteristic impedance
formula can be written in the following format:

Zo = sqrt ( (R + 2 * pi * f * L ) / (G + j * 2 * pi * f * c) )

Where:

R = The series resistance of the conductor in ohms per unit length (DC resistance)

G = The shunt conductance in mhos per unit length

j = A symbol indicating that the term has a phase angle of +90 degres (imaginary number)

pi = 3.1416

L = Cable inductance per unit lenght

C = Cable capacitance per unit lenght

sqrt = square root function

For materials commonly used for cable insulation, G is small
enough that it can be neglected when compared with 2(3.1416) f C.
At low frequencies, 2(3.1416) f L is so small compared with R that
it can be neglected. Therefore, at low frequencies, the following
equation can be used:

Zo = sqrt ( R / (j * 2 * pi * f * C))

If the capacitance does not vary with frequency, the Zo varies inversely with the square root of the frequency and has a phase angle which is -45o near DC and decreases to 0o as frequency increases. Polyvinyl chloride and rubber decrease somewhat in capacitance as frequency increases, while polyethylene, polypropylene, and Teflon* do not vary significantly.

When f becomes large enough, the two terms containing f become so large that R and G may be neglected and the resultant equation is:

Zo = sqrt ( (j * 2 * pi * f * L) / (j * 2 * pi * f * C) )

Which can be simplified to the form:

Zo = sqrt ( L / C )

Cables characteristics at high frequencies

At the high frequencies you can't look at the cable as a usual cable. On
higher frequency it works as a waveguide. Characteristic impedance is
specific resistance for electro-magnetic waves.
So: It's the load the cable poses at high frequencies. Thehe high
frequency goes (dependent of cable of course) usually from 100kHz and
up.

If you feed a sinusoidal electrical AC signal of reasonable frequency
into one end of the cable, then the signal travels as an electrical wave
down the cable. If the cable length is an extremely large number of
wave-lengths at the frequency of that AC signal, and you measure the
ratio of AC Voltage to AC current in that traveling wave, then that ratio is
called the characteristic impedance of the cable.

In practical cables the characteristic impedance is determined by
cable geometry and dielectric. The cable length has no effect of it's
characteristic impedance.

What does the coaxial cable model look like ?

The coax is represented schematically by a series of
capacitors and inductances, a sort of odd filter arrangement, the
particular values unique to the particular coax type. At a given
frequency, if correctly chosen, that arrangement passes most of the signal;
while at higher frequencies, that arrangment attentuates signal.

How does coaxial cable chacteristics define the impedance ?

The length has nothing to do with a coaxial cable impedance.
Characteristic impedance is determined by the size and spacing of the
conductors and the type of dielectric used between them.
For ordinary coaxial cable used at reasonable frequency, the
characteristic impedance depends on the dimensions of the inner and
outer conductors, and on the characteristics of the dielectric material
between the inner and outer conductors.

The following formula can be used
for calculating the characteristic impedance of the coaxial cable:
(formula taken from Reference Data for Radio Engineers book published
by Howard W. Sams & Co. 1975, page 24-21)

impedance = (138 / e^(1/2)) * log (D/d)

Where:

log = logarithm of 10

d = diameter of center conductor

D = inner diameter of cable shield

e = dielectric constant (= 1 for air)

In a nut shell the characteristic impedance of a coax cable is the
square root of (the per unit length inductance divide by the per unit
length capacitance).
For coaxial cables the characteristic impedance will be typically
between 20 and 150 ohms. The length of
the cable makes no difference whatsoever in regard to the characteristic impedance.

If the frequency is much too high for the coaxial
cable, then the wave can propagate in undesired modes (i.e., have
undesired patterns of electric and magnetic fields), and then the cable
does not function properly for various reasons.

How is the impadance of balanced pairs ?

Characteristic impedance is determined by the size and spacing of the
conductors and the type of dielectric used between them.
Balanced pair, or twin lines, have a Zo which depends on the ratio of the
wire spacing to wire diameter and the foregoing remarks still apply.
For practical lines, Zo at high frequencies is very nearly, but not exactly,
a pure resistance.

The following formula can be used
for calculating the characteristic impedance of balanced pair near ground:
(formula taken from Reference Data for Radio Engineers book published
by Howard W. Sams & Co. 1975, page 24-22)

impedance = (276 / e^(1/2)) * log ((2D/d) * (1 + (D/2h)^2))^(1/2))

Where:

log = logarithm of 10

d = wire diameter

D = distance between wires in pair

e = dielectric constant (= 1 for air)

h = distance between balanced pair and ground

Not that this formula is only valid for unshielded balanced pair
when D and h are order of magnitude larger than d.
If the twisted pair is far away from ground (h is nearly infinite),
the the effect of the ground is neglegtible and
the impedance of the cable can be approximated with simpler formula
(my own derivation from formula above):

impedance = (276 / e^(1/2)) * log ((2D/d)

For twin line Zo will be typically between 75 and 1000 ohms depending on
the intended application. The impedance of typical old telephone pair
in the telephone poles in the air has characteristic impedance of
around 600 ohms. The telephone and telecommunication cables in use
have typically a characteristic impedance of 100 or 120 ohms.

What kind of electrical model I can use for long coaxial cable ?

If you know the imductance and capacitance of certain lenght of
cable you can use the following electrical model for it:

Since the chain is infinite, the terms on the right are just equal to Z. You get a nice quadratic.

"long" isn't real restrictive so as to be in the wavelength or better ballpark.

Can I measure the cable impedance using multimeter ?

Cable characteristic impedance is a cable characteristics which
is only valid for high frequency signals. Multimeters use DC current
for resistance measurements, so you can't measure the cable impedance
using your multimeter or other simple measurement equipments.
It is usually best to check the cable type (usually printed on cable)
and it's characteristics impedance from some catalogue instead of
trying to measure it.

How can I measure cable impedance ?

A relationship exists which makes determination of Zo rather simple with the proper equipment. It can be shown that if, at a given frequency, the impedance of a length of cable is measure with the far end open (Zoc), and the measurement is repeated with the far end shorted (Zsc), the following equation may be used to determine Zo:

Zo = sqrt ( Zoc * Zsc )

Where:

Zoc = impedance of a length of cable is measure with the far end open

Zsc = impedance of a length of cable is measure with the far end shorted

NOTE: The Zoc and Zsc measurements both have magnitude and phase, so the Zo will also have magnitude and phase.

High frequency measurements of Zo are made by determining the velocity of propagation and capacitance of the cable or by reflectometry.

When cable impedance effects the signal ?

In order for a cable's characteristic impedance to make any
difference in the way the signal passes through it, the cable must be
at least a large fraction of a wavelength long for the particular
frequency it is carrying.

Most wires will have a speed of travel for AC current of 60 to 70 percent of
the speed of light, or about 195 million meters per second.
An audio frequency of 20,000 Hz has a
wavelength of 9,750 meters, so a cable would have to be four or five
*kilometers* long before it even began to have an effect on an audio
frequency. That's why the characteristic impedance of audio interconnect
cables is not something most of us have anything to worry about.

Normal video signal rarely exceed 10 MHz. That's about
20 meters for a wavelength. Those frequencies are getting close to being
high enough for the characteristic impedance to be a factor.
High resolution computer video signals and fast digita signals
easily exceed 100 MHz so the proper impedance matching is needed
even in shor cable runs.

How impedance matching works

First, you want to drive the cable with
an electrical source that has an output impedance equal to the
characteristic impedance of the cable,
so that all of the source's output power goes into
the cable, rather than being reflected from the cable's input end back
into the source. Second, you want the electrical load on the output end
of the cable, to have an input impedance equal to the characteristic
impedance of the cable, so that all of the power goes into the load
rather than being reflected from the load back into the cable.

There are many exceptions to this normal driving method, but those
are used for for special effects. You can choose an impedance match for
maximum power transfer at low bandwidth, or mismatch the impedance for
a flatter frequency response. It's the engineer's call, depending on what
he wants.

Why impedance matching is needed ?

If you have mismatches between the source's output impedance,
the cable's characteristic impedance, and load's input impedance, then
the reflections can depend critically on the length of the cable. And if
you distort the cable, as by crushing or kinking, or if you install
connectors improperly, then you can have reflections, with resulting power
loss. And
sometimes reflected power can damage the power source if lits of power
is sent to the cable (e.g., a radio transmitter).
So you need to be careful of impedance mismatches.

An anomoly that is not in all text books is when antenna pushes
power back (not a proper termination), it looks at the inside
of the shield and the outside, which ever one is lowest gets the power .
This means the RF can travel on the outside of the coax.
The most difficult concept about coax is the XL,XC do not exist
(to your transmitter) if cable is terminated.

Most common reasons for listing a cable impedance is that because
of its reliable electrical characteristics, and that very impedance
listing. Coax is often is used to carry low level higher frequency signals
that are separated. Separations are very expensive in terms of signal loss
-- a perfect impedance match will cost you half the signal, and even a
slight mismatch is very costly, particularly at antenna strength signals.
Carefully matched carriers, like coax, are necessary to preserve signal at
reduced noise.

What effect does the nominal capacitance have on the cable's
performance or transmission capabilities?

Capacitance of the cable is nothing to do if the coax is terminated.
The transmitter will see absolutely no capacitance nor inductance.

And this transmission line characteristic is used to hide capacitance in
high frequnecy PCB's. Engineers can design the PCB traces so that they have
the proper capacitance and inductance values so that the transmitter will
see nothing but a proper impedance transmission line.

Why is characteristic impedance important in data transmission?

If a cable is terminated in its matching characteristic impedance you can't tell from the sending end that the cable is not infinitely long - all the signal that is fed into the cable is taken by the cable and the load.

If the impedances are not matches, part of the waves in the cable
will be reflected back on the cable connections distorting the outbound waves.
When these reflected waves hit the wave generator, they are again reflected and mingle with the outbound waves so that it is difficult to tell which waves are original and which are re-reflections.

The same thing happens when pulses are sent down the cable - when they encounter an impedance other than the characteristic impedance of the cable, a portion of their energy is reflected back to the sending end. If the pulses encounter an open circuit or a short circuit, all of the energy is reflected (except for losses due to attenuation - another subject). For other terminations, smaller amounts of energy will be reflected.

This reflected energy distorts the pulse, and if the impedance of the pulse generator is not the same as the characteristic impedance of the cable, the energy will be re-reflected back down the cable, appearing as extra pulses.

Can I use coaxial cable without impedance matching ?

If the coaxial cable is very short, the cable impedance does not
have much effect on the signal. Usually the beast way to transmit
signal through coaxial cable is to do the impedance matching,
although there are some applications where the normal impedance
matching on both ends is not done. In some special applications
the cable might be only impedance matched at only one end
or intentionally mistached at both ends. Those application
are special cases, where the cable impedance is take acount
so that the combination of the cable and the terminations
at the ends of the cable produce the desired transmission
characteristics to the whole system. In this kind of
special application the cable is not considered as a
passive transmission line, but a signal modifying component
in the circuit.

What about the velocity of propogation ratio ?

Velocity of propogation ratio percentage based on the speed of light in
vacuum. The percentage tells what is the speed of the signal in the cable
compared to the speed of light in vacuum.
In coax cable, under reasonable conditions, the propagation velocity
depends on the characteristics of the dielectric material.

Why attenuation figures tend to increase with increasing frequency ?

That usually is due mainly to the limited penetration of current into
the inner and outer conductors (the skin effect). With increasing
frequency, the current penetrates less deeply into the conductors, and
thus is confined to a thinner region of metal. Therefore the
resistance, hence attenuation, is higher. It also can be caused partly
by energy loss in the dielectric material.

How to minimize the attenuation in coax ?

For a line with fixed outer conductor diameter, and whose outer
and inner conductors have the same resistivity, and assuming
you use a dielectric with negligible loss (such as polyethylene
or Teflon in the high-frequency range at least), then you get
minimum loss in coax if you minimize the expression:

(1/d + 1)/ln(1/d)

where d is the ratio of inner conductor diameter to outer
conductor ID.
A spreadsheet or calculator gets you close
pretty quickly: D/d = 3.5911 is close.
Thr formula was claimed to be derived from the formula
for coax impedance versus D/d and a formula for loss that you'll
find in "Reference Data for Engineers" published by Howard Sams,
on pg. 29-13 in the seventh edition.

The interesting thing to
notice is that this minimum loss does not directly yield a
line impedance: the line impedance depends on the dielectric
constant of the dielectric. For air insulated line, the
corresponding impedance is about 76.71 ohms, but if the line
is insulated with solid polyethylene, then minimum attenuation
is at about 50.6 ohms. So, however it came to be, all the
RG-58 we use for antenna feeds and test equipment connections
is pretty close to minimum attenuation given the above
conditions, and that the dielectric is polyethylene.

But if the line uses foam dielectric with a velocity factor of 0.8,
then the impedance of minimum atten would be about 61 ohms.
However, that minimum is a pretty broad one, and you don't
start loosing a lot till you get more than perhaps 50% away
from the optimal impedance.

Note that foam-dielectric line with the same impedance and
outer diameter as solid-dielectric line will have lower loss.
That's because, to get the same impedance, the foam line
will have a larger inner conductor, and that larger conductor
will have lower RF resistance, and therefore lower loss.

Typical cable impedances

What are typical cable impedances ?

The most typical coaxial cable impedances used are 50 and 75 ohm coaxial
cables. 50 ohm coaxial cables might be the most commonly used coaxial
cables and they are used commonly with radio transmitters, radio
receivers, laboratory equipments and in ethernet network.

Another commonly used cable type is 75 ohm ciaxial cable which is
used in video applications, in CATV networks, in TV antenna wiring
and in telecommunication applications.

600 ohms is a typical impednace for open-wire balanced lines for
telegraphy and telephony.
A twisted pairs of 22 gage wire with
reasonable insulation on the wires comes out at about 120 ohms for the
same mechanical reasons that the other types of transmission lines
have their own characteristic impedances.

Twin lead used in some antenna systema are 300 ohms to match to a
folded dipole in free space impedance (However, when that folded dipole
is part of a Yagi (beam) antenna, the impedance is usually quite a bit
lower, in the 100-200 ohm range typically.).

Why 50 ohm coax ?

Standard coaxial line impedance for r.f. power transmission in
the U.S. is almost exclusively 50 ohms. Why this value was chosen is
given in a paper presented by _Bird Electronic Corp._
Standard coaxial line impedance for r.f. power transmission in
the U.S. is almost exclusively 50 ohms. Why this value was chosen is
given in a paper presented by Bird Electronic Corp.

Different impedance values are optimum for different parameters.
Maximum power-carrying capability occurs at a diameter ratio of 1.65
corresponding to 30-ohms impedance. Optimum diameter ratio for voltage
breakdown is 2.7 corresponding to 60-ohms impedance (incidentally, the
standard impedance in many European countries).

Power carrying capacity on breakdown ignores current density
which is high at low impedances such as 30 ohms. Attenuation due to
conductor losses alone is almost 50% higher at that impedance than at
the minimum attenuation impedance of 77 ohms (diameter ratio 3.6).
This ratio, however, is limited to only one half maximum power of a
30-ohm line.

In the early days, microwave power was hard to come by and lines
could not be taxed to capacity. Therefore low attenuation was the
overriding factor leading to the selection of 77 (or 75) ohms as a
standard. This resulted in hardware of certain fixed dimensions. When
low-loss dielectric materials made the flexible line practical, the
line dimensions remained unchanged to permit mating with existing
equipment.

The dielectric constant of polyethylene is 2.3. Impedance of a
77-ohm air line is reduced to 51 ohms when filled with polyethylene.
Fifty-one ohms is still in use today though the standard for precision
is 50 ohms.

The attenuation is minimum at 77 ohms; the breakdown voltage is maximum at 60 ohms
and the power-carrying capacity is maximum at 30 ohms.

Another thing which might have lead to 50 ohm coax is that
if you take a reasonable sized center conductor and put a
insulator around that and then put a shield around that and choose all
the dimensions so that they are convenient and mechanically look good,
then the impedance will come out at about 50 ohms. In order to raise
the impedance, the center conductor's diameter needs to be tiny with
respect to the overall cable's size. And in order to lower the
impedance, the thickness of the insulation between the inner conductor
and the shield must be made very thin.
Since almost any coax that *looks* good for mechanical reasons
just happens to come out at close to 50 ohms anyway, there was a
natural tendency for standardization at exactly 50 ohms.

Cable capacitance and characteristic impedance

Take a chunk of coax, connected to nothing. The center conductor and
shield form a capacitor. If you charge that capacitor up to 100V,
then short the shield to the center conductor, What is the current
flow?

It is not infinite (or determined by parasitic resistance anc
reactance ) like a "normal capacitor" but it is determined by the
characteristic impedance of the line. If it is 50 ohm line charged to
100V then the current WILL be 2Amps. (100/50) It will be a square
pulse, and temporal width (time duration, pulse width whatever you
choose to call it) will be determined by the length of the
line (around 1.5 nS/foot depending on line's velocity factor).

This method can be used for example to generate current pulses to
semicondictor lasers. To get the pulse lengths longer than easily availabe
with practical coaxial lines you can use use lumped impedance near-equivilant.

Using coaxial cables in applications

What happens if I use 50 ohm cable for vidoe application which needs 75 ohm cable ?

If 50 ohm cable sees a 75 ohm load (the receiver), a substantial part of the
signal will be reflected back to the transmitter. Since the transmitter is also 75 ohm, this
relected signal will be substantially reflected back to the receiver. Because
of the delay, it will show up as a nasty ghost in the picture. Multiple ghosts
like this look like ringing. Also, the reflections cause partial signal
cancellations at various frequencies.

How can I convert cable impedance values ?

The cable impedance itself can't be converter unless you replace
the whole cable with new one which has the right impedance.
If you absolutelu need to use the existing cable for your application
then there is one way to use the exiting cable: impedance converters.
There are transformers which can make the cable look like different impedance
cable when those are installed to both ends of the cable.

In some application it is possible to resistive adapers to convert
the cable impedances. Those adapters are simpler than transformers but
typically have a noticable signal loss in them (typically around 6 db
for 75 ohm to 50 ohm conversion).

Impedance of circuit board traces

High speed signals can be routed on a circuit board if care is taken
to make the impedance of the traces match the source driver impedance
and the destination termination impedance. A microstrip line will
exhibit a characteristic impedance if the thickness, width, and height
of the line above the ground plane are controlled.

The dielectric constant, Er, for typical 0.062" fiberglass board is
4.8. Using a trace thickness of 0.00134" gives a line width of 109
mils for a 50 ohm microstrip.

When routing circuit board traces, differential pairs should have the
same length trace. These trace lines should also be as short as possible.

Impedance matching between different impedances

If two cables with different impedances are connected
togerther or a cable is connected to a source which has different
impedance then some kind of impdance matching is needed to
avoid the signal reflections in the place where the cables
are connected together.

Using transformer for impedance matching

The most classical method for matching different impedances is
to use a matching transformer with proper impedance tranfer ratio.
The impednace tranfer ratio of a transformer is determined by
using the formula:

Za / (Na^2) = Zb / (Nb^2)

Where:

Za = input impedance

Na = number of turns on input coil

Zb = output impedance

Nb = number of turns on output coil

The equation can be converted to format:

Zb = Za * (Nb/Na)^2

From that equation you can see that Nb/Na is same as the
transformer voltage transferrign ratio between
primaty and secondary. This means that when you know
that ratio you can use the equation without knowing
the exact turns ratio.

Impedance matching netweork usign resistors

The matching network shown below can be used to match two
unequal impedances, provided that Z1 is grater than Z2.

As you can see from the table the cost of simple resistor
based impedance matching is quite large signal level attenuation
in the conversion process.

Comments

I have received the following comment on the cable impedance equation:

I have read your document, which is I must say very well written.
I found a small mistake there though in "How does coaxial cable chacteristics define the impedance ?".
Your formula is *impedance = (138 / e^(1/2)) * log (D/d)*, but this is only true for ideal conductor.
Speed of wave in copper is less than in vacuum, and it equals to about 248827740 m/s; this means that it should be multiplied by a factor of 0.83.
So the formula should look like this: